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Re: Sum of Infinite Series



I don't think this is a math question, it's a physics question.
When the cg of the pile is outside of the base of the pile then the stack
will tumble.
Many a ship was lost at sea by ignoring (a variation of) this
principle.
This is a sophisticated version of the "what color were the
engineer's suspenders" kind of question.
Regards,
Jack
On Thu, 24 Jan 2002, Paul O. Johnson wrote:

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This is a math question. Please bear with me. I can't find the answer in any of my few physics and math reference books. Since it's based on center of gravity and stability principles, I feel emboldened to ask it here.

I am creating a tabletop exhibit at The Science Place. It consists of eight identical rectangular wooden blocks (about 8 inches long). The object is to create an "impossible stairway" by stacking the blocks so that the top block extends beyond the supporting edge of the bottom block. It looks unstable but it ain't.

The theoretical best solution is to stack the blocks from the top down so that the steps get shorter as you work your way down. The top step (between blocks 1 and 2) is 1/2 block-length, the next step down (between blocks 2 and 3) is 1/4 block-length, then 1/6, 1/8, 1/10, 1/12, and finally 1/14 block-length between blocks 7 and 8.

I want to state in the exhibit's sign what the absolute maximum extension is. This requires that I sum the infinite series 1/2 + 1/4 + 1/6 + 1/8 + . . . I'm ashamed to admit that I no longer carry this trivial math equation in my head (I know I once did). Can anyone help me?

Paul O. Johnson
Exhibit Developer
The Science Place
Dallas, Texas

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<DIV><FONT face=Arial size=2>This is a math question. Please bear with me. I
can't find the answer in any of my few physics and math reference books. Since
it's based on center of gravity and stability principles, I feel emboldened to
ask it here.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>I am creating a tabletop exhibit at The Science
Place. It consists of eight identical rectangular wooden blocks (about 8 inches
long). The object is to create an "impossible stairway" by stacking the blocks
so that the top block extends beyond the supporting edge of the bottom block. It
looks unstable but it ain't.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>The theoretical best solution is to stack the
blocks from the top down so that the steps get shorter as you work your way
down. The top step (between blocks 1 and 2) is 1/2 block-length, the next step
down (between blocks 2 and 3) is 1/4 block-length, then 1/6, 1/8, 1/10, 1/12,
and finally 1/14 block-length between blocks 7 and 8.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>I want to state in the exhibit's sign what the
absolute maximum extension is. This requires that I sum the infinite series 1/2
+ 1/4 + 1/6 + 1/8 + . . . I'm ashamed to admit that I no longer carry this
trivial math equation in my head (I know I once did). Can anyone help
me?</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT>&nbsp;</DIV>
<DIV><FONT face=Arial size=2>Paul O. Johnson</FONT></DIV>
<DIV><FONT face=Arial size=2>Exhibit Developer</FONT></DIV>
<DIV><FONT face=Arial size=2>The Science Place</FONT></DIV>
<DIV><FONT face=Arial size=2>Dallas, Texas</FONT></DIV></BODY></HTML>

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--
"But as much as I love and respect you, I will beat you and I will kill
you, because that is what I must do. Tonight it is only you and me, fish.
It is your strength against my intelligence. It is a veritable potpourri
of metaphor, every nuance of which is fraught with meaning."
Greg Nagan from "The Old Man and the Sea" in
<The 5-MINUTE ILIAD and Other Classics>